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© 2026 Physics Playground. Feito para os curiosos.Idioma

Preface
1Functions
2Linear Functions
3Polynomial and Rational Functions
4Exponential and Logarithmic Functions
5Trigonometric Functions
6Periodic Functions
7Trigonometric Identities and Equations
8Further Applications of Trigonometry
9Systems of Equations and Inequalities
10Analytic Geometry
11Sequences, Probability and Counting Theory
12Introduction to Calculus
ABasic Functions and Identities
Answer Key
Index

Biblioteca Precalculus

Sumário

Preface
1Functions
2Linear Functions
3Polynomial and Rational Functions
4Exponential and Logarithmic Functions
5Trigonometric Functions
6Periodic Functions
7Trigonometric Identities and Equations
8Further Applications of Trigonometry
9Systems of Equations and Inequalities
10Analytic Geometry
11Sequences, Probability and Counting Theory
12Introduction to Calculus
ABasic Functions and Identities
Answer Key
Index

A

Basic Functions and Identities

Appendix A Basic Functions and Identities

Graphs of the Parent Functions

Three graphs side-by-side. From left to right, graph of the identify function, square function, and square root function. All three graphs extend from -4 to 4 on each axis.

Figure A1

Three graphs side-by-side. From left to right, graph of the cubic function, cube root function, and reciprocal function. All three graphs extend from -4 to 4 on each axis.

Figure A2

Three graphs side-by-side. From left to right, graph of the absolute value function, exponential function, and natural logarithm function. All three graphs extend from -4 to 4 on each axis.

Figure A3

Graphs of the Trigonometric Functions

Three graphs of trigonometric functions side-by-side. From left to right, graph of the sine function, cosine function, and tangent function. Graphs of the sine and cosine functions extend from negative two pi to two pi on the x-axis and two to negative two on the y-axis. Graph of tangent extends from negative pi to pi on the x-axis and four to negative 4 on the y-axis.

Figure A4

Three graphs of trigonometric functions side-by-side. From left to right, graph of the cosecant function, secant function, and cotangent function. Graphs of the cosecant function and secant function extend from negative two pi to two pi on the x-axis and ten to negative ten on the y-axis. Graph of cotangent extends from negative two pi to two pi on the x-axis and twenty-five to negative twenty-five on the y-axis.

Figure A5

Three graphs of trigonometric functions side-by-side. From left to right, graph of the inverse sine function, inverse cosine function, and inverse tangent function. Graphs of the inverse sine and inverse tangent extend from negative pi over two to pi over two on the x-axis and pi over two to negative pi over two on the y-axis. Graph of inverse cosine extends from negative pi over two to pi on the x-axis and pi to negative pi over two on the y-axis.

Figure A6

Three graphs of trigonometric functions side-by-side. From left to right, graph of the inverse cosecant function, inverse secant function, and inverse cotangent function.

Figure A7

Trigonometric Identities

Pythagorean Identities	$\begin{array}{l} \cos^{2} θ + \sin^{2} θ = 1 \\ 1 + \tan^{2} θ = \sec^{2} θ \\ 1 + \cot^{2} θ = \csc^{2} θ \end{array} \begin{array}{l} \cos^{2} θ + \sin^{2} θ = 1 \\ 1 + \tan^{2} θ = \sec^{2} θ \\ 1 + \cot^{2} θ = \csc^{2} θ \end{array}$
Even-Odd Identities	$\begin{array}{l} \cos (- θ) = \cos θ \\ \sec (- θ) = \sec θ \\ \sin (- θ) = - \sin θ \\ \tan (- θ) = - \tan θ \\ \csc (- θ) = - \csc θ \\ \cot (- θ) = - \cot θ \end{array} \begin{array}{l} \cos (- θ) = \cos \\ \sec (- θ) = \sec \\ \sin (- θ) = - \sin \\ \tan (- θ) = - \tan \\ \csc (- θ) = - \csc \\ \cot (- θ) = - \cot \end{array}$
Cofunction Identities	$\begin{array}{l} \cos θ = \sin (\frac{π}{2} - θ) \\ \sin θ = \cos (\frac{π}{2} - θ) \\ \tan θ = \cot (\frac{π}{2} - θ) \\ \cot θ = \tan (\frac{π}{2} - θ) \\ \sec θ = \csc (\frac{π}{2} - θ) \\ \csc θ = \sec (\frac{π}{2} - θ) \end{array} \begin{array}{l} \cos \\ \sin \\ \tan \\ \cot \\ \sec \\ \csc \end{array}$
Fundamental Identities	$\begin{array}{l} \tan θ = \frac{\sin θ}{\cos θ} \\ \sec θ = \frac{1}{\cos θ} \\ \csc θ = \frac{1}{\sin θ} \\ cot θ = \frac{1}{tan θ} = \frac{cos θ}{sin θ} \end{array} \begin{array}{l} \tan \\ \sec \\ \csc \\ cot \end{array}$
Sum and Difference Identities	$\begin{array}{l} \cos (α + β) = \cos α \cos β - \sin α \sin β \\ \cos (α - β) = \cos α \cos β + \sin α \sin β \\ \sin (α + β) = \sin α \cos β + \cos α \sin β \\ \sin (α - β) = \sin α \cos β - \cos α \sin β \\ \tan (α + β) = \frac{\tan α + \tan β}{1 - \tan α \tan β} \\ \tan (α - β) = \frac{\tan α - \tan β}{1 + \tan α \tan β} \end{array} \begin{array}{l} \cos (α + β) = \cos \\ \cos (α - β) = \cos \\ \sin (α + β) = \sin \\ \sin (α - β) = \sin \\ \tan (α + β) = \frac{\tan}{1 - \tan} \\ \tan (α - β) = \frac{\tan}{1 + \tan} \end{array}$
Double-Angle Formulas	$\begin{array}{l} \sin (2 θ) = 2 \sin θ \cos θ \\ \cos (2 θ) = \cos^{2} θ - \sin^{2} θ \\ \cos (2 θ) = 1 - 2 \sin^{2} θ \\ \cos (2 θ) = 2 \cos^{2} θ - 1 \\ \tan (2 θ) = \frac{2 \tan θ}{1 - \tan^{2} θ} \end{array} \begin{array}{l} \sin (2 θ) = 2 \\ \cos (2 θ) = \cos^{2} θ - \sin^{2} θ \\ \cos (2 θ) = 1 - 2 \\ \cos (2 θ) = 2 \\ \tan (2 θ) = \frac{2}{1 - \tan^{2} θ} \end{array}$
Half-Angle Formulas	$\begin{array}{l} \sin \frac{α}{2} = \pm \sqrt{\frac{1 - \cos α}{2}} \\ \cos \frac{α}{2} = \pm \sqrt{\frac{1 + \cos α}{2}} \\ \tan \frac{α}{2} = \pm \sqrt{\frac{1 - \cos α}{1 + \cos α}} \\ \tan \frac{α}{2} = \frac{\sin α}{1 + \cos α} \\ \tan \frac{α}{2} = \frac{1 - \cos α}{\sin α} \end{array} \begin{array}{l} \sin \\ \cos \\ \tan \\ \tan \\ \tan \end{array}$
Reduction Formulas	$\begin{array}{l} \sin^{2} θ = \frac{1 - \cos (2 θ)}{2} \\ \cos^{2} θ = \frac{1 + \cos (2 θ)}{2} \\ \tan^{2} θ = \frac{1 - \cos (2 θ)}{1 + \cos (2 θ)} \end{array} \begin{array}{l} \sin^{2} θ = \frac{1 - \cos (2 θ)}{2} \\ \cos^{2} θ = \frac{1 + \cos (2 θ)}{2} \\ \tan^{2} θ = \frac{1 - \cos (2 θ)}{1 + \cos (2 θ)} \end{array}$
Product-to-Sum Formulas	$\begin{array}{l} \cos α \cos β = \frac{1}{2} [\cos (α - β) + \cos (α + β)] \\ \sin α \cos β = \frac{1}{2} [\sin (α + β) + \sin (α - β)] \\ \sin α \sin β = \frac{1}{2} [\cos (α - β) - \cos (α + β)] \\ \cos α \sin β = \frac{1}{2} [\sin (α + β) - \sin (α - β)] \end{array} \begin{array}{l} \cos \\ \sin \\ \sin \\ \cos \end{array}$
Sum-to-Product Formulas	$\begin{array}{l} \sin α + \sin β = 2 \sin (\frac{α + β}{2}) \cos (\frac{α - β}{2}) \\ \sin α - \sin β = 2 \sin (\frac{α - β}{2}) \cos (\frac{α + β}{2}) \\ \cos α - \cos β = - 2 \sin (\frac{α + β}{2}) \sin (\frac{α - β}{2}) \\ \cos α + \cos β = 2 \cos (\frac{α + β}{2}) \cos (\frac{α - β}{2}) \end{array} \begin{array}{l} \sin \\ \sin \\ \cos \\ \cos \end{array}$
Law of Sines	$\begin{array}{l} \frac{\sin α}{a} = \frac{\sin β}{b} = \frac{\sin γ}{c} \\ \frac{a}{\sin α} = \frac{b}{\sin β} = \frac{c}{\sin γ} \end{array} \begin{array}{l} \frac{\sin}{a} = \frac{\sin}{b} = \frac{\sin}{c} \\ \frac{a}{\sin} = \frac{b}{\sin} = \frac{c}{\sin} \end{array}$
Law of Cosines	$\begin{array}{l} a^{2} = b^{2} + c^{2} - 2 b c \cos α \\ b^{2} = a^{2} + c^{2} - 2 a c \cos β \\ c^{2} = a^{2} + b^{2} - 2 a b cos γ \end{array} \begin{array}{l} a^{2} = b^{2} + c^{2} - 2 b c \\ b^{2} = a^{2} + c^{2} - 2 a c \\ c^{2} = a^{2} + b^{2} - 2 a b \end{array}$

Table A1

A Basic Functions and Identities · Precalculus · Physics Playground

Key Equations

Key Equations

Exercises

Review Exercises
Practice Test

Exercises

Review Exercises
Practice Test

Zeros of Polynomial Functions

Zeros of Polynomial Functions

3.7Rational Functions

3.7Rational Functions

3.8Inverses and Radical Functions

3.8Inverses and Radical Functions

3.9Modeling Using Variation

3.9Modeling Using Variation

Chapter Review

Key Terms
Key Equations
Key Concepts

Chapter Review

Key Terms
Key Equations
Key Concepts

Exercises

Review Exercises
Practice Test

Exercises

Review Exercises
Practice Test

4.6

4.6

Exponential and Logarithmic Equations

Exponential and Logarithmic Equations

4.7Exponential and Logarithmic Models

4.7Exponential and Logarithmic Models

4.8Fitting Exponential Models to Data

4.8Fitting Exponential Models to Data

Chapter Review

Key Terms
Key Equations
Key Concepts

Chapter Review

Key Terms
Key Equations
Key Concepts

Exercises

Review Exercises
Practice Test

Exercises

Review Exercises
Practice Test

Exercises

Review Exercises
Practice Test

Exercises

Review Exercises
Practice Test

Key Concepts

Key Concepts

Exercises

Review Exercises
Practice Test

Exercises

Review Exercises
Practice Test

7.6

7.6

Modeling with Trigonometric Functions

Modeling with Trigonometric Functions

Chapter Review

Key Terms
Key Equations
Key Concepts

Chapter Review

Key Terms
Key Equations
Key Concepts

Exercises

Review Exercises
Practice Test

Exercises

Review Exercises
Practice Test

8.6

8.6

Parametric Equations

Parametric Equations

8.7Parametric Equations: Graphs

8.7Parametric Equations: Graphs

Chapter Review

Key Terms
Key Equations
Key Concepts

Chapter Review

Key Terms
Key Equations
Key Concepts

Exercises

Review Exercises
Practice Test

Exercises

Review Exercises
Practice Test

9.6

9.6

Solving Systems with Gaussian Elimination

Solving Systems with Gaussian Elimination

9.7Solving Systems with Inverses

9.7Solving Systems with Inverses

9.8Solving Systems with Cramer's Rule

9.8Solving Systems with Cramer's Rule

Chapter Review

Key Terms
Key Equations
Key Concepts

Chapter Review

Key Terms
Key Equations
Key Concepts

Exercises

Review Exercises
Practice Test

Exercises

Review Exercises
Practice Test

Key Terms

Key Terms

Exercises

Review Exercises
Practice Test

Exercises

Review Exercises
Practice Test

11.6

11.6

Binomial Theorem

Binomial Theorem

11.7Probability

11.7Probability

Chapter Review

Key Terms
Key Equations
Key Concepts

Chapter Review

Key Terms
Key Equations
Key Concepts

Exercises

Review Exercises
Practice Test

Exercises

Review Exercises
Practice Test

Key Equations

Key Equations

Exercises

Review Exercises
Practice Test

Exercises

Review Exercises
Practice Test

Chapter 9

Chapter 9

AnteriorPractice Test Próxima Chapter 1

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